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Instructional principles gleaned from cognitive science play a critical role in improving classroom teaching. This study examines how three cognitive instructional principles including worked examp...

This study examines how sampled Chinese and U.S. third and fourth grade students (NChina = 167, NUS = 97) understand the commutative, associative, and distributive properties. These students took b...

This study reports how an expert Chinese teacher implements mathematics textbook lessons in enacted instruction. Our video analysis indicates that both textbook and enacted teaching included only one worked example; however, the teacher engaged students in unpacking the example in great depth. Both the textbook and the enacted teaching showed “concreteness fading” in students’ use of representations. However, the Chinese teacher incorporated students’ self-generated representations and facilitated students’ active modeling of quantitative relationships. Finally, the Chinese teacher asked a greater number of deep questions than were suggested by the textbook. These deep questions often occurred as clusters of follow-up questions that were either concept-specific or promoted comparisons which facilitated connection-making between multiple representations and solutions.

Two teachers used a powerful, challenging tool in their Chinese classroom to build, ensure, and solidify students' understanding of quantitative relationships.

This study explores how preservice teachers (PSTs) transfer the intended specialized content knowledge (SCK) to elementary classrooms. Focusing on the case of the associative property of multiplica...

In this study we investigate the teaching of the associative property in a natural classroom setting through observation of classroom video of several elementary math classes in a large urban schoo...

Prior studies show that elementary school children generally “lack” formal understanding of inverse relations. This study goes beyond lack to explore what children might “have” in their existing conception. A total of 281 students, kindergarten to third grade, were recruited to respond to a questionnaire that involved both contextual and non-contextual tasks on inverse relations, requiring both computational and explanatory skills. Results showed that children demonstrated better performance in computation than explanation. However, many students’ explanations indicated that they did not necessarily utilize inverse relations for computation. Rather, they appeared to possess partial understanding, as evidenced by their use of part-whole structure, which is a key to understanding inverse relations. A close inspection of children’s solution strategies further revealed that the sophistication of children’s conception of part-whole structure varied in representation use and unknown quantity recognition, which suggests rich opportunities to develop students’ understanding of inverse relations in lower elementary classrooms.

This study, focusing on inverse relations, examines how representative U.S. and Chinese elementary textbooks may provide opportunities to learn fundamental mathematical ideas. Findings from this study indicate that both of the U.S. textbook series (grades K-6) in comparison to the Chinese textbook samples (grades 1–6), presented more instances of inverse relations, while also containing more unique types of problems; yet, the Chinese textbooks provided more opportunities supporting meaningful and explicit learning. In particular, before presenting corresponding practice problems, Chinese textbooks contextualized worked examples of inverse relations in real-world situations to aid in sense making of computational or checking procedures. The Chinese worked examples also differed in representation uses especially through concreteness fading. Finally, the Chinese textbooks spaced learning over time, systematically stressing structural relations including the inverse quantities relationships. These fin...

Researchers have long debated the meaning of mathematical understanding and ways to achieve mathematical understanding. This study investigated experienced Chinese mathematics teachers’ views about mathematical understanding. It was found that these mathematics teachers embrace the view that understanding is a web of connections, which is a result of continuous connection making. However, in contrast to the popular view which separates understanding into conceptual and procedural, Chinese teachers prefer to view understanding in terms of concepts and procedures. They place more stress on the process of concept development, which is viewed as a source of students’ failures in transfer. To achieve mathematical understanding, the Chinese teachers emphasize strategies such as reinventing a concept, verbalizing a concept, and using examples and comparisons for analogical reasoning. These findings draw on the perspective of classroom practitioners to inform the long-debated issue of the meaning of mathematical understanding and ways to achieve mathematical understanding.

Through examining a representative Chinese textbook series’ presentation of the distributive property, this study explores how mathematics curriculum may structure representations in ways that facilitate the transition from concrete to abstract so as to support students’ learning of mathematical principles. A total of 319 instances of the distributive property were identified. The representational transition among these instances was analyzed at three tiers: within one worked example, from the worked example to practice problems within one topic, and across multiple topics over grades. Findings revealed four features that facilitate the transition process in the Chinese textbook series. First, it situates initial learning in a word problem context, which serves as a starting point of the transition process. Second, it sets up abstract representations as an ultimate goal of the multi-tier transition process. Third, it incorporates problem variations with connections in carefully designed tasks that embody the same targeted principles. Fourth, it engages students in constant sense making of the transition process through various pedagogical supports. Implementations and future research directions are also discussed.

This study explores, from both constructivist and cognitive perspectives, teacher guidance in student-centered classrooms when addressing a common learning difficulty with equivalent fractions—lines or pieces—based on number line models. Findings from three contrasting cases reveal differences in teachers’ facilitating and direct guidance in terms of anticipating and responding to student difficulties, which leads to differences in students’ exploration opportunity and quality. These findings demonstrate the plausibility and benefit of integrating facilitating and direct guidance in student-centered classrooms. Findings also suggest two key components of effective teacher guidance including (a) using pretraining through worked examples and (b) focusing on the relevant information and explanations of concepts. Implementations are discussed.

The purpose of this cross-national study is to understand teachers’ views about the meaning of instructional coherence and the ways to achieve instructional coherence. With respect to the meaning of instructional coherence, whereas the majority of U.S. teachers paid attention to connections between teaching activities, lessons, or topics, the majority of Chinese teachers emphasized the interconnected nature of mathematical knowledge beyond the teaching flow. U.S. teachers expressed their views about ways to achieve instructional coherence through managing a complete lesson structure. In contrast, Chinese teachers emphasized pre-design of teaching sequences, transitional language and questioning based on the study of textbooks and students beforehand. Moreover, they emphasized addressing student thinking and dealing with emerging events in order to achieve “real” coherence. The findings of the study contribute to our understanding about the meaning of instructional coherence and ways to achieve instructional coherence in different cultural contexts.

This study examines preservice elementary teachers’ (PTs) knowledge for teaching the associative property (AP) of multiplication. Results reveal that PTs hold a common misconception between the AP and commutative property (CP). Most PTs in our sample were unable to use concrete contexts (e.g., pictorial representations and word problems) to illustrate AP of multiplication conceptually, particularly due to a fragile understanding of the meaning of multiplication. The study also revealed that the textbooks used by PTs at both the university and elementary levels do not provide conceptual support for teaching AP of multiplication. Implications of findings are discussed.

Specialized content knowledge (SCK) is a type of mathematical content knowledge specifically needed for teaching. This type of knowledge, although serving as a critical component for preservice teacher education, is often challenging to develop with preservice elementary teachers (PTs). The purpose of this study is to investigate PTs’ development of SCK for teaching fundamental mathematical ideas in a university methods course. Focusing on the case of the associative property (AP) of multiplication, the author as the course instructor identified three instructional opportunities (a formal introduction and two delayed revisits) to stress two SCK components, representations, and explanations. PTs’ learning progresses were assessed through three diagnostic tests (a pretest, a mid-term exam, and a final exam) and two prompts, which informed the upcoming lesson design. Meanwhile, the course instructor conducted ongoing reflections on PTs’ learning, which also informed the corresponding lessons. It was found that PTs initially generated abstract number sentences without reasoning about the contexts of word problems. This representational sequence indicates a symbol precedence view. When prompted for explanations, PTs focused on individual numbers rather than quantitative relationships, and they could not consistently apply the basic meaning of multiplication for reasoning. The methods course, when designed to address these issues, promoted PTs’ SCK development. At the end of the course, the majority of PTs were able to generate number sentences based on the word problem structures and provided reasonable explanations; however, the methods course also faced dilemmas due to PTs’ robust symbol precedence view and the tension between PTs’ learning and children’s learning. Very few studies have explored ways to support PTs’ knowledge growth in SCK, especially for teaching fundamental mathematical ideas. This study, by carefully documenting the successes and challenges in developing PTs’ SCK, contributes to the existing literature. Based on our findings, this study highlights the importance of stressing basic meanings so as to develop PTs’ explanation skills. Meanwhile, to develop PTs’ representation skills, university instructors should be aware of the tension between PTs’ and children’s learning as manifested by PTs’ symbol precedence view. Finally, to support PTs’ SCK growth, it is also important to emphasize the role of elementary textbooks.

This study took a broad view of generalization, exploring how classroom instruction within opportunities of implicit generalization may be related to students’ abilities to generalize. From an integrated perspective of generalization and problem solving, we investigated twelve lessons of two middle level teachers based on three pattern-based problems. We found teachers’ limited awareness and knowledge of problem situations, especially problem structures, were obstacles hindering their ability to support students’ understanding of implicit generalization. Teachers tended to focus on particular "results" for individual cases rather than the "process" of obtaining those results, resulting in missed opportunities to develop students’ sense of general patterns. Without attention to implicit generalization, explicit generalization rarely took place in a meaningful way during problem solving.

This study qualitatively explored teachers’ responses to doubling errors (e.g., 3/4 × 2 = 6/8) that typically reflect students’ difficulties in understanding the “rule” for finding equivalent fractions (e.g., 3/4 × 2/2 = 6/8). Although all teachers claimed to teach for understanding in interviews, their responses varied in terms of effectiveness and types. One teacher grasped the doubling error to explain the reasons underlying the rule using symbolic representations only. The other two used concrete representations to explicate the reasons but tended to either prevent or ignore the doubling error. Although these teaching actions appeared different in form, all teachers essentially held the same perception-based perspectives; that is, they either assumed students could see what they saw or students could directly comprehend mathematical ideas through visualizing pictures only. Such perspectives overlook students’ available mathematical conceptions and might be at the root of the difficulties with ...

In this study, we investigated Chinese teachers' attributions and coping strategies for classroom misbehaviour across grade levels. A total of 244 teachers (Grades 1–12) from the Chinese mainland participated in this survey. Results indicated that Chinese teachers first attributed misbehaviour to student characteristics, such as being “lazy, not making enough effort”, and second to “bad learning habits”. Looking across grade levels, elementary teachers first blamed student learning habits while secondary teachers blamed student effort. With regard to coping strategies, inconsistencies were found across grade levels and between teachers' perceptions and actions. The majority of sampled elementary teachers tended to choose “praising good students” as the most effective and often-used strategy, while secondary teachers believed in “talking after class”. In fact, “talking after class” was viewed to be more effective as grade level increased. However, teachers reported that they did not actually use the strate...

This study examines presentations of the distributive property (DP) in two widely used U.S. elementary text series and one main Chinese text series along three dimensions: problem contexts, typical problem types within each problem context, and variability in using the DP. In general, the two U.S. texts were found to resemble each other but to differ considerably from the Chinese text series. Both U.S. texts are computation-dominated, presenting various strategies centering on “breaking apart a factor to perform multiplication.” These strategies limit the use of the DP mainly with whole numbers and in a regular direction. The underlying principle of these strategies is seldom made explicit. In contrast, the Chinese text approaches focus on the underlying principle and are well aligned with cognitive research suggestions. Multiple-step word problems with particular structures are used in a systematic and hierarchic manner across grades to help students learn and transfer the DP. The Chinese texts also tend...

As a novel generation of a rotational gas wave machine, the wave rotor refrigerator (WRR) is an unsteady flow device used for refrigeration, in whose passages pressured streams directly contact and exchange energy due to the movement of pressure waves. In this paper, the working mechanism and refrigeration principle are investigated based on the one-dimensional unsteady flow theory. A basic limitation on main structural parameters and operating parameters is deduced and the wave diagram of WRR to guide designing is sketched. The main influential factors are studied through an experiment. In the DUT Gas Wave Refrigeration Studying and Development Center (GWRSDC) lab, the isentropic efficiency can now reach about 65%. The results show that the WRR is a feasible and promising technology in pressured gas refrigeration cases.

This study reports findings from comparative samples of sixth-grade Chinese and U.S. students' interpretations of the equal sign. Ninety-eight percent of the Chinese sample correctly answered 4 items indicating conceptions of equality and provided conceptually accurate explanations. In contrast, only 28% of the U.S. sample performed at this level. We examine how teacher preparation materials, students' textbooks and teachers' guidebooks treat equality in each country. U.S. teacher preparation textbooks rarely interpreted the equal sign as equivalence. On the contrary, Chinese textbooks typically introduced the equal sign in a context of relationships and interpreted the sign as “balance,” “sameness,” or “equivalence” and only then embedded the sign with operations on numbers.

This study focuses on Chinese teachers' perceptions of students' classroom misbehaviour. A questionnaire was designed to assess teachers' general concerns about classroom management, teachers' perceptions of the most frequent and troublesome types of misbehaviour, and teachers' perceived needs for help with improving classroom management. A total of 244 responses were collected, from lower elementary to high school teachers in two provinces in mainland China. In general, the majority of Chinese teachers (65.6%) do not think that classroom management is a great concern. They perceive ‘daydreaming’ to be the most frequent and troublesome misbehaviour. This contrasts with many prior studies in Western settings where ‘talking out of turn’ has been reported as the biggest concern. In addition, Chinese teachers are eager to understand the psychological reasons behind student misbehaviour. They also suggest that the Chinese system of ‘teaching for testing’ should be reformed.

The authors examined the extent to which teacher interventions focused on students' mathematical thinking in naturalistic cooperative-learning mathematics classroom settings. The authors also observed 6 videotapes about the same teaching content using similar curriculum from 2 states. They created 2 instruments for coding the quality of teacher intervention length, choice and frequency, and intervention. The results show the differences of teacher interventions to improve students' cognitive performance. The authors explained how to balance peer resource and students' independent thinking and how to use peer resource to improve students' thinking. Finally, the authors suggest detailed techniques to address students' thinking, such as identify, diversify, and deepen their thinking.

This study examined students' conceptions about the equal sign in light of historical findings with an international comparison group. Textbooks for preparation of students as mathematics teachers were examined. Participants were sixth-grade students from the USA ( n = 105) and China ( n = 145). About 98% of the Chinese children correctly answered all items by providing conceptually accurate explanations, but only 28% of the U.S. sample did. Textbooks for education majors who would teach in the USA rarely discussed the equal sign as equivalence while the Chinese texts introduced the “equal sign” in a context of relationships and discussed it as “balance,” “sameness,” or “equivalence.”